Maximum Likelihood Estimation Example

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I am doing MAB final year and making project report came to heard about the term Maximum likelihood estimation (MLE) wants to know about this. Will you tell basic details about Maximum likelihood estimation (MLE) also provide Maximum Likelihood Estimation Example ?

[sumit]

Maximum likelihood estimation (MLE) is a method of estimating the parameters of a statistical model, given observations.

MLE attempts to find the parameter values that maximize the likelihood function, given the observations.


Maximum Likelihood Estimation Example






Steps for Maximum Likelihood Estimation

The above discussion can be summarized by the following steps:

Start with a sample of independent random variables X1, X2, . . . Xn from a common distribution each with probability density function f(x;θ1, . . .θk). The thetas are unknown parameters.
Since our sample is independent, the probability of obtaining the specific sample that we observe is found by multiplying our probabilities together. This gives us a likelihood function L(1, . . .k) = f( x1 ;1, . . .k) f( x2 ;1, . . .k) . . . f( xn ;1, . . .k) = f( xi ;1, . . .k).
Next we use Calculus to find the values of theta that maximize our likelihood function L.

More specifically, we differentiate the likelihood function L with respect to θ if there is a single parameter. If there are multiple parameters we calculate partial derivatives of L with respect to each of the theta parameters.
To continue the process of maximization, set the derivative of L (or partial derivatives) equal to zero and solve for theta.

We can then use other techniques (such as a second derivative test) to verify that we have found a maximum for our likelihood function.


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